First plot the field, by the foregoing rules; then divide it into triangles, and measure their bases and perpendiculars; then work as follows.

137

33.

Note-You must always use the same line on the scale, to find the content, that you used in plotting.

TO LAY OUT LAND.

To lay out any number of aeres in a square figure.
RULE.

Annex to the number of acres given, 5 cyphers, which will turn the acres into links, extract the square root from the number thus increased, which will be the side of the proposed square.

It is required to lay out 100 acres in a square figure. I join to the 100 five cyphers, and then it is 100,00000 square links; the root of which is 3162 nearest, or 31 ch. 62 lin. the length of one side of the square.

To lay out any number of acres in form of a Parallelogram, whereof one side is given. RULE.

First turn the acres into links by annexing five cyphers; di vide that number by the given side, the quotient will be the oth er side.

It is required to lay out 100 acres in a parallelogram, one side of which shall be 20 chains; to the 100 I annex five oyphers, and it is 109,00000, which divided by 20 chains, or 2000 links, gives 50 chains for the other side of the parallelogram

To lay out any number of acres in form of a triangle upon a given base. RULE.

To double the number of acres, annex five cyphers, and divide by the base, the quotient will be the length of the perpendicular required.

It is required to lay out 100 acres in form of a triangle upon a given base, whose length is 40 chains.

To double the 100 acres, I annex five cyphers, which make 200,00000, which, divided by 40, gives 50 chains for the height of the perpendicular.

To find the length of the diameter of a cirele, which shall contain any number of acres required:

RULE.

As 11 is to 14: so is the number of acres given to the square of the diameter, of the circle, required.

It is required to lay out 100 acres in form of a circle; the length of the diameter is required.

Add five cyphers to the 100, and it makes 100.00000; then, as 11 14: 100,00000: 12727272; the root of which is 35 ch. 67 or 68 lin. the diameter of the circle required.

OF DIVIDING LANDS.

To divide a triangle several ways.

Suppose A B C contain 60 acres, to be divided between two men, one to have 40 acres cut off towards A, ande other 20

Multiply and divide, the quotient will be 33 ch. 33 lin.; which set off upon the base from A to D, and draw the line B D, which shall divide the triangle as was required.

To divide a triangle into any number of unequal parts, by lines proceeding from any à assigned point.

The triangle A B C contains 60 acres, to be divided between 3 men; the first to have 15 acres, the second 20, and the third 25 acres; the lines of division to proceed from D_First measure the base, which is 50 chains; then as 60: 50: : 15: 12 ch. 50 lin. to be set off from A to E, for the first man's base : Abi 60: 50: : 20: 16 ch: 66 lin. which set off from E to then, consequently, the third sinks. This done

F, for the second man's

man's base viz from F to C, must be 20 chains, or

draw a line from the
point assigned, D,to the
opposite angle B and
from E and F draw the
lines E H, FG parallel'
to B D. Lastly, from D
draw the lines D G, D
11 which divide the tri-
angle into three such A
parts as were required.

To divide a piece of land, in form of a Triangle, according to any proportion given, by a line parallel to one of the sides.

The triangle ABC

contains 60 acres,the base AC is 50 chams. It is required to divide it, by a line parallel to B C. into two parts; the one to contain 40 acres, the other 20.

Divide the base as be- A

fore taught, and at the line

of division fall in D; A

B

20

E

D being 3 chains, 33 links, and D C 16 chains, 67 links. Multiply the hole base 50 by A D 33 chains, 33 links, the product is 1666,500 extract the square root, and it gives 40 chains, 82 links set t off from A to E. From E draw E F parallel te BC, which vides the triangle as required. To cut off from

a square any assigned part," laying in form of a par

allelogram,

RULE.

Extract the square root of the number of rods contained in the square; divide the number of rods, to be cut off by the root, the quotient will be the breadth of the side to be taken off.

The square A B CD contains 34 A. 3 R. 20 rods; and it is required to cut off 8 A. 2 R. 35 rods

The 34 A. 3 R. 20 rods reduced to rods, are 5580, the square root of which is 74,60 equal to one de of the square. Then the 8 A. 2 R. 35 rods, reduced to rods, are 1395; which divided by 749 give 18,6

E

A

rods, to be set off

',;

B

C

F

D

from A to E, and from D to F. Then will the parallelogram A EFD contain 8 A. 2 R. 35 rods.

To divide a circle according to any proportion, by a line concentric with the first

The areas of circles are in proportion to one another as the squares of their semidiameters therefore if you divide the square of the semidiameter by the proportion given, and extract the square root, you will have your desire.

Let A CBD be a circle to be equally di vided between two men. The diameter. thereof is 2 chains. The semidiameter 1 eh. or 100 links. The square thereof is 10000. Halfthesquare A is 5000. The foot of the half is 71 links, nearly; which take from your scale, and upon the same centre describe the circle G EHF, which will divide the circle A B C Dinto two equalparts.

C

TO MEASURE DISTANCES GEOMETRICALLY If you were at A, and were de. sirous of knowing the distance from A to C ; but by reason of a river or some other obstacle, you could not measure it, you might in the following manner, obtain the true distance.

With a compass, or any other instrument used in taking angles, placed at A, observe what point C bears from you, which let be S. 73045. then turn your compass to wards some other object, as B; note the bearing and distance, which let be S. 30 rods. Then remove your compass to B, and note the bearing to C, which let be N. 67 E.

A

739

30 rods,

67°

Now to find the distance, draw a line A B, representing a south line, and from any scale of equal parts, lay thereon 30 from A to B then lay your protractor on the point at A, and set off 73°, drawing a line from A through that point; then lay your protractor on the point at B, and set off 67o, drawing an other line intersecting the former; the meeting of these two. lines in C completes the triangle; then taking the line AC in your compasses, and applying it to the same line of equal parts, it will give the distance required, 45 rods.

Note The ground between your two stations, as A and B, should be level.

TO MEASURE HEIGHTS GEOMETRICALLY.. Let A B represent a tree or building, standing perpendicu